JEE Main & Advanced Mathematics Mathematical Logic and Boolean Algebra Basic Logical Connectives or Logical Operators

Definition : The phrases or words which connect simple statements are called logical connectives or sentential connectives or simply connectives or logical operators.

In the following table, we list some possible connectives, their symbols and the nature of the compound statement formed by them. 

(i) Conjunction : Any two simple statements can be connected by the word “and” to form a compound statement called the conjunction of the original statements.

Symbolically if p and q are two simple statements, then \[p\wedge q\] denotes the conjunction of p and q and is read as “p and q”.

(ii) Disjunction or alternation Any two statements can be connected by the word “or” to form a compound statement called the disjunction of the original statements.

Symbolically, if p and q are two simple statements, then \[p\vee q\] denotes the disjunction of p and q and is read as “ p or q”.

(iii) Negation : The denial of a statement p is called its negation, written as ~ p.

Negation of any statement p is formed by writing “ It is not the case that ..... “ or “ It is false that.......” before p or, if possible by inserting in p the word “not”.

• Negation is called a connective although it does not combine two or more statements. In fact, it only modifies a statement.

(iv) Implication or conditional statements : Any two statements connected by the connective phrase “if.. then” give rise to a compound statement which is known as an implication or a conditional statement.

If p and q are two statements forming the implication ‘if p then q¢, then we denote this implication by

\[''p\Rightarrow q''\,\,\text{or}\,\,''p\to q''\].

In the implication \[''p\Rightarrow q'',\,p\]is the antecedent and q is the consequent.

Truth table for a conditional a statement  

(v) Biconditional statement : A statement is a biconditional statement if it is the conjunction of two conditional statements (implications) one converse to the other.

Thus, if p and q are two statements, then the compound statement  \[p\Rightarrow q\] and \[q\Rightarrow p\] is called a biconditional statements or an equivalence and is denoted by \[p\Leftrightarrow q\].

Thus, \[p\Leftrightarrow q\] : \[(p\Rightarrow q)\wedge (q\Rightarrow p)\]

Truth table for a biconditional statement : Since \[p\Leftrightarrow q\] is the conjunction of  \[p\Rightarrow q\] and \[q\Rightarrow p\]. So, we have the following truth table for \[p\Leftrightarrow q\].